Question 357978: n one version of the game Keno, the house has a pot containing 80 balls, numbered 1 through 80. A player buys a ticket for $1 and marks one number on it (from 1 through 80). The house then selects 20 of the 80 numbers at random. If the number selected by the player is among the 20 selected by the house (the management), the player is paid $3.20. Find the expected net winnings for this game.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
The probability of catching exactly  spots when you have bet on  out of 80 spots is given by:
\ =\ \left(N\cr \,r\right\)\ \cdot\ \frac{\left(80\,-\,N\cr 20\,-\ r\right\)}{\left(80\cr 20\right\)})
Where ) is the number of combinations of  things taken  at a time and is calculated by !})
And the expected payout for a bet on out of 80 spots is given by:
\ \cdot\ \frac{\left(80\,-\,N\cr 20\,-\ i\right\)}{\left(80\cr 20\right\)}W_i)
where  is the payout for catching  spots when you have bet on  out of 80 spots and  is the amount bet.
For your problem, you only have to count from 0 to 1, your amount bet is $1, and the payout for 1 out of 1 is $3.20, so:
\ \cdot\ \frac{\left(80\,-\,1\cr 20\,-\,0\right\)}{\left(80\cr 20\right\)}\ \cdot\ 0\ +\ \left(1\cr 1\right\)\ \cdot\ \frac{\left(80\,-\,1\cr 20\,-\,1\right\)}{\left(80\cr 20\right\)}\ \cdot\ 3.20)
Of course, since there is zero payout for not hitting the number, this reduces to:
\ \cdot\ \frac{\left(80\,-\,1\cr 20\,-\,1\right\)}{\left(80\cr 20\right\)}\ \cdot\ 3.20)
Still, you are faced with what looks like a nasty bunch of arithmetic.
But because we know that \ =\ 1\ \forall\ n\ \in\ \mathbb{N}) , this becomes:
\ \cdot\ \frac{\left(80\,-\,1\cr 20\,-\,1\right\)}{\left(80\cr 20\right\)}\ =\ \frac{\frac{79!}{60!19!}}{\frac{80!}{60!20!}}\ =\ \left(\frac{79!}{60!19!}\right)\left(\frac{60!20!}{80!}\right))
And because we know from the definition of factorials !) , we can write:
\ \cdot\ \frac{\left(80\,-\,1\cr 20\,-\,1\right\)}{\left(80\cr 20\right\)}\ =\ \frac{20}{80}\ =\ \frac{1}{4})
And finally:
This is actually a big payout. Most Vegas casinos get 25 cents from every dollar played.
John
My calculator said it, I believe it, that settles it
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